Question

Two cylindrical wires of identical length are made of copper and aluminum. If they carry the same current and have the same potential difference across their length, what is the ratio of their radii?

Short answer

Answer: The approximate ratio of the radii of the copper and aluminum wires is 1.296.

Step-by-step solution

1. Write the resistance formula for each wire

For the copper wire, the resistance is given by: R_Cu = ρ_Cu * L / A_Cu For the aluminum wire, the resistance is given by: R_Al = ρ_Al * L / A_Al

2. Express the area in terms of radii

Both wires are cylindrical, so the cross-sectional area is given by A = πr^2. A_Cu = π * r_Cu^2 A_Al = π * r_Al^2 Thus, we can rewrite the resistance formulas as follows: R_Cu = ρ_Cu * L / (π * r_Cu^2) R_Al = ρ_Al * L / (π * r_Al^2)

3. Set the resistances equal to each other

Since both wires have the same current and potential difference, their resistances are equal. We can equate the above formulas: ρ_Cu * L / (π * r_Cu^2) = ρ_Al * L / (π * r_Al^2)

4. Simplify and solve for the ratio of radii

We can cancel out the L and π terms from both sides and rearrange the equation to find the ratio of their radii: r_Cu^2 / r_Al^2 = ρ_Al / ρ_Cu Take the square root of both sides: r_Cu / r_Al = sqrt(ρ_Al / ρ_Cu)

5. Calculate the ratio using known resistivity values

Now, we can plug in the known resistivity values for copper (ρ_Cu = 1.68 × 10^-8 Ω⋅m) and aluminum (ρ_Al = 2.82 × 10^-8 Ω⋅m) to find the ratio of their radii: r_Cu / r_Al = sqrt((2.82 × 10^-8 Ω⋅m) / (1.68 × 10^-8 Ω⋅m)) r_Cu / r_Al ≈ sqrt(1.679) ≈ 1.296 So, the ratio of the radii of the copper and aluminum wires is approximately 1.296.